A character approach to directed genus distribution of graphs: The bipartite single-black-vertex case

Abstract

Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a positive answer for these families to a question asked in Bonnington et al. (2002) [1]. Our proof uses real-rooted polynomials and the representation theory of the symmetric group Sn. The result is also extended to some factorizations of the identity in Sn that are rotation systems of some families of one-face constellations.